Sound Waves in Deformed Perfectly Elastic Materials. Acoustoelastic Effect

TLDR

The paper compares necessary and sufficient conditions for plane wave propagation in deformed materials with restrictions from static equilibrium. The study aims to analyze small‑amplitude plane waves in initially deformed, stressed elastic materials. The authors derive general equations for small displacements on finite deformations and compatibility conditions linking sound wave data to classical elasticity theory for arbitrary material symmetry. The work determines restrictions on the finite elasticity energy function and demonstrates that sound speed variations with stress can be used to extract third‑order elastic constants, providing experimental validation.

Abstract

The general equations for a small displacement superimposed on a finite deformation of a perfectly elastic material of arbitrary symmetry are derived anew. These equations are the basis of an analysis of plane waves of small amplitude propagating in an initially deformed and stressed elastic material. Certain restrictions on the energy function of finite elasticity theory are determined. These restrictions, which provide necessary and sufficient conditions that a homogeneously deformed material admit plane waves, are then compared with other restrictions obtained from themostatic equilibrium considerations. Compatibility conditions, which are necessary and sufficient that data on sound wave propagation in elastic materials be compatible with classical elasticity theory upon suitable assignment of the material symmetry, are derived. Finally, it is shown how the variation of sound speeds with initial stress and the measured magnitude of the acoustoelastic effect can be used to determine the third-order elastic constants of an isotropic material and as a partial confirmation and experimental check of the theory.